An Application of Optimal Interpolation Formula with Derivative to Approximate Integration
DOI:
https://doi.org/10.71310/pcam.2_72.2026.10Keywords:
interpolation, spline, quadrature formula, integration, approximationAbstract
In this paper, an optimal interpolation formula with derivative constructed in the Sobolev space is considered. The formula interpolates an unknown function using its values and derivatives first three order at equally spaced nodes. Explicit expressions for the coefficients of the corresponding quadrature formula are derived by integrating the interpolation basis functions. A theorem giving the exact form of the coefficients is presented. Numerical experiments are carried out for several smooth functions, and the absolute errors of the approximate integration are analyzed for different values of ????. The results show that the proposed approach provides high accuracy and can be effectively used for numerical integration problems where derivative information is available. In addition, the stability of the proposed formula with respect to perturbations in the input data is examined, and its asymptotic behavior as the number of nodes increases is discussed. Comparisons with classical interpolation and quadrature formulas demonstrate the advantage of incorporating derivative information, especially for highly smooth functions. The method also provides a constructive framework for extending optimal interpolation formulas to higher-order derivatives and nonuniform meshes. These results contribute to the broader development of optimal computational schemes in Sobolev spaces and highlight potential applications in solving boundary value problems, differential equations, and numerical simulation tasks requiring high precision.
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